\section{Conclusions and Future Work}
\label{sec:discussions}

In this paper, we proposed and developed a SAT-based framework for
cost sensitive temporally expressive (CSTE) planning, an important
but difficult planning domain that is potentially useful for many
applications.
Our work was motivated by the observations that high
action concurrency is a main characteristic of temporally expressive
planning problems and that it is often desired to optimize action costs.
Such high concurrency and cost sensitivity were exemplified by the new
P2P communication network domain we introduced in this paper.

We proposed a general framework for CSTE planning which translates a CSTE planning problem into a MinCost SAT problem, an
optimization problem with SAT-clause constraints.
%Two approaches for solving MinCost SAT problems were developed under
%this framework. One is to use a generic Max-SAT solver, and the other
We used a branch-and-bound procedure based on the DPLL algorithm to solve the MinCost SAT problem.
We developed two planning-specific schemes to improve the branch-and-bound
procedure, including a relaxed-planning based lower bounding
mechanism for stronger node pruning, and an action-cost based scheme
for variable branching.

Our experimental results in several CSTE domains showed that our new solver is able to find solutions
with the minimum makespans and low action costs. We showed
that the proposed schemes are very effective in improving search
efficiency. The proposed planners compare favorably against
existing temporally expressive planners.

Comparing to heuristic search based approaches, our framework, as an anytime algorithm, enjoys at least two advantages.
The first advantage is its flexibility. Given more time, our new planner can
always search for better solutions with lower total action costs. 
%Second, we can leverage the extensive research on
%SAT. We can always expect performance enhancements, by adapting more
%efficient Max-SAT or MinCost SAT solvers.
%In addition, we can easily
%utilize additional SAT instance evaluation
%strategies~\cite{Rintanen06:AIJ} to further improve the overall performance.
The second advantage is its capability in handling
high concurrencies. Our results show that, in those problems with
high concurrencies, our framework has clear advantages over existing
temporally expressive planners.

It is an continuing effort to advance the state-of-the-art
of temporally expressive planning.  While we developed an efficient CSTE
planner with encouraging experimental results in several problem
domains, our work can be further extended. One limitation of our
current method is that it is not expressive enough for numerical
constraints and other complex PDDL2.1 properties. For example,
comparing to Crikey, the major limitation of our current
implementation is that it does not support \emph{variable} action
durations.
To fully support PDDL2.1 and to take advantage of SAT-based
planning techniques, we plan to enhance our planner to handle
more complex temporal constraints and other richer semantics.

%Furthermore, although we presented the P2P network domain as a case
%study of an application, in the current formulation, we
%ignored some technically involved, practical issues, such as the
%cost for establishing a connection, the potential of data
%compression that might take additional processing time but shorten
%communication delay, file segmentation to further increase
%concurrency, and stochastic nature of communication channels.
%We plan to augment the P2P planning domain by incorporating these features.
